Problem: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{x^2 - 36}{x + 6}$
Explanation: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = x$ $ b = \sqrt{36} = 6$ So we can rewrite the expression as: $q = \dfrac{({x} + {6})({x} {-6})} {x + 6} $ We can divide the numerator and denominator by $(x + 6)$ on condition that $x \neq -6$ Therefore $q = x - 6; x \neq -6$